Subgroups — C₂×C₃×#3155 #10679248

Order 768.

3534 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#535	C₂	1	2	yes	maximal, complemented
5	C₃×#3155	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃×#496	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×#495	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#493	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#3155	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×#41	C₂²	1	2	yes	—
12	C₂³×C₃×D₄	C₂²	1	2	yes	—
13	C₂×C₃×AGL(1,ℤ/8)	—	2	16	no	—
14	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	16	yes	complemented
15	C₂×C₃×#135	—	2	8	no	—
16	C₂×C₃×#135	C₂²	1	8	yes	complemented
17	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃×SD₁₆	C₂²	1	4	yes	complemented
19	C₂²×C₃×SD₁₆	C₂²	1	2	yes	complemented
20	C₂²×C₃×D₈	—	2	8	no	—
21	C₂²×C₃×D₈	C₂²	1	2	yes	—
22	C₂²×C₃×D₈	C₂²	1	2	yes	—
23	C₂²×C₃×D₈	C₂²	1	8	yes	complemented
24	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₃×D₈	C₂²	1	4	yes	complemented
26	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
27	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
28	C₃×#535	—	2	8	no	—
29	C₃×#535	C₂²	1	16	yes	complemented
30	C₂×C₃×#127	C₂²	1	4	yes	complemented
31	C₂×C₃×#124	C₂²	1	4	yes	complemented
32	C₂×C₃×#121	—	2	4	no	—
33	C₂×C₃×#120	—	2	4	no	—
34	C₂×C₃×#117	C₂²	1	1	yes	characteristic
35	C₃×#496	C₂²	1	32	yes	complemented
36	C₃×#495	C₂²	1	16	yes	complemented
37	C₃×#493	C₂²	1	16	yes	complemented
38	C₃×#405	C₂²	1	8	yes	complemented
39	C₂×C₃×#97	C₂²	1	2	yes	complemented
40	C₂×C₃×C₄×C₈	C₂²	1	2	yes	complemented
41	C₂²×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
42	C₂²×#135	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂³×D₈	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×#535	C₂×C₃	1	2	yes	complemented
45	#3155	C₂×C₃	1	16	yes	direct product factor
46	C₂×#496	C₂×C₃	1	4	yes	complemented
47	C₂×#495	C₂×C₃	1	2	yes	complemented
48	C₂×#493	C₂×C₃	1	2	yes	complemented
49	C₂×#405	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁵×C₃	—	2	2	no	—
51	C₂×C₃×#41	—	2	4	no	—
52	C₂×C₃×#41	—	2	8	no	—
53	C₂×C₃×#41	—	2	8	no	—
54	C₂×C₃×#41	C₂³	1	8	yes	—
55	C₂²×C₃×Q₈	C₂³	1	2	yes	—
56	C₂²×C₃×D₄	—	2	2	no	—
57	C₂²×C₃×D₄	—	2	8	no	—
58	C₂²×C₃×D₄	—	2	8	no	—
59	C₂²×C₃×D₄	—	2	2	no	—
60	C₂²×C₃×D₄	—	2	4	no	—
61	C₂²×C₃×D₄	—	2	8	no	—
62	C₂²×C₃×D₄	C₂³	1	4	yes	—
63	C₂²×C₃×D₄	C₂³	1	2	yes	—
64	C₂²×C₃×D₄	C₂³	1	8	yes	—
65	C₂³×C₃×C₄	—	2	2	no	—
66	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₃×AGL(1,ℤ/8)	—	2	64	no	—
68	C₃×#135	—	2	32	no	—
69	C₂×C₃×Q₁₆	—	2	4	no	—
70	C₂×C₃×Q₁₆	C₂³	1	4	yes	complemented
71	C₂×C₃×SD₁₆	—	2	8	no	—
72	C₂×C₃×SD₁₆	—	2	16	no	—
73	C₂×C₃×SD₁₆	C₂³	1	8	yes	complemented
74	C₂×C₃×SD₁₆	C₂³	1	16	yes	complemented
75	C₂×C₃×D₈	—	2	8	no	—
76	C₂×C₃×D₈	—	2	8	no	—
77	C₂×C₃×D₈	—	2	4	no	—
78	C₂×C₃×D₈	—	2	16	no	—
79	C₂×C₃×D₈	—	2	32	no	—
80	C₂×C₃×D₈	C₂³	1	8	yes	—
81	C₂×C₃×D₈	C₂³	1	8	yes	—
82	C₂×C₃×D₈	C₂³	1	4	yes	complemented
83	C₂×C₃×D₈	C₂³	1	16	yes	complemented
84	C₂×C₃×M₁₆	D₄	1	8	yes	complemented
85	C₂×C₃×M₁₆	C₂³	1	4	yes	—
86	C₂²×C₃×C₈	D₄	1	8	yes	complemented
87	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
88	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
89	C₂²×C₃×C₈	C₂³	1	2	yes	—
90	C₂²×C₃×C₈	C₂³	1	4	yes	—
91	C₃×#127	—	2	8	no	—
92	C₃×#127	C₂³	1	16	yes	complemented
93	C₃×#124	—	2	8	no	—
94	C₃×#124	C₂³	1	16	yes	complemented
95	C₃×#121	—	2	32	no	—
96	C₃×#120	—	2	32	no	—
97	C₃×#117	D₄	1	4	yes	—
98	C₃×#117	C₂³	1	4	yes	—
99	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
100	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
101	C₂×C₃×C₂²⋊C₄	D₄	1	2	yes	—
102	C₂×C₃×C₄²	C₂³	1	2	yes	—
103	C₃×#97	C₂³	1	8	yes	complemented
104	C₃×C₄×C₈	C₂³	1	8	yes	complemented
105	C₂²×#41	C₂²×C₃	1	2	yes	—
106	C₂³×D₄	C₂²×C₃	1	2	yes	—
107	C₂×AGL(1,ℤ/8)	—	2	16	no	—
108	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	16	yes	complemented
109	C₂×#135	—	2	8	no	—
110	C₂×#135	C₂²×C₃	1	8	yes	complemented
111	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
112	C₂²×SD₁₆	C₂²×C₃	1	4	yes	complemented
113	C₂²×SD₁₆	C₂²×C₃	1	2	yes	complemented
114	C₂²×D₈	—	2	8	no	—
115	C₂²×D₈	C₂²×C₃	1	2	yes	—
116	C₂²×D₈	C₂²×C₃	1	2	yes	—
117	C₂²×D₈	C₂²×C₃	1	8	yes	complemented
118	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₂²×D₈	C₂²×C₃	1	4	yes	complemented
120	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
121	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
122	#535	—	2	8	no	—
123	#535	C₂²×C₃	1	16	yes	complemented
124	C₂×#127	C₂²×C₃	1	4	yes	complemented
125	C₂×#124	C₂²×C₃	1	4	yes	complemented
126	C₂×#121	—	2	4	no	—
127	C₂×#120	—	2	4	no	—
128	C₂×#117	C₂²×C₃	1	1	yes	characteristic
129	#496	C₂²×C₃	1	32	yes	complemented
130	#495	C₂²×C₃	1	16	yes	complemented
131	#493	C₂²×C₃	1	16	yes	complemented
132	#405	C₂²×C₃	1	8	yes	complemented
133	C₂×#97	C₂²×C₃	1	2	yes	complemented
134	C₂×C₄×C₈	C₂²×C₃	1	2	yes	complemented
135	C₂⁴×C₃	—	2	4	no	—
136	C₂⁴×C₃	—	2	2	no	—
137	C₂⁴×C₃	—	2	8	no	—
138	C₂⁴×C₃	—	4	8	no	—
139	C₂⁴×C₃	—	8	8	no	—
140	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
141	C₃×#41	—	2	16	no	—
142	C₃×#41	—	2	16	no	—
143	C₃×#41	—	4	16	no	—
144	C₂×C₃×Q₈	—	2	2	no	—
145	C₂×C₃×Q₈	—	2	4	no	—
146	C₂×C₃×Q₈	—	2	4	no	—
147	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
148	C₂×C₃×D₄	—	2	8	no	—
149	C₂×C₃×D₄	—	2	8	no	—
150	C₂×C₃×D₄	—	2	16	no	—
151	C₂×C₃×D₄	—	2	4	no	—
152	C₂×C₃×D₄	—	2	8	no	—
153	C₂×C₃×D₄	—	2	4	no	—
154	C₂×C₃×D₄	—	2	16	no	—
155	C₂×C₃×D₄	—	2	8	no	—
156	C₂×C₃×D₄	—	2	4	no	—
157	C₂×C₃×D₄	—	2	2	no	—
158	C₂×C₃×D₄	—	2	32	no	—
159	C₂×C₃×D₄	—	4	8	no	—
160	C₂×C₃×D₄	—	4	8	no	—
161	C₂×C₃×D₄	—	4	16	no	—
162	C₂×C₃×D₄	—	8	16	no	—
163	C₂×C₃×D₄	C₂⁴	1	8	yes	—
164	C₂×C₃×D₄	C₂⁴	1	4	yes	—
165	C₂²×C₃×C₄	—	2	2	no	—
166	C₂²×C₃×C₄	—	2	8	no	—
167	C₂²×C₃×C₄	—	4	8	no	—
168	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
169	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
170	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
172	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
173	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
174	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
175	C₃×Q₁₆	—	2	8	no	—
176	C₃×SD₁₆	—	2	16	no	—
177	C₃×SD₁₆	—	2	32	no	—
178	C₃×D₈	—	2	16	no	—
179	C₃×D₈	—	2	16	no	—
180	C₃×D₈	—	2	8	no	—
181	C₃×D₈	—	2	32	no	—
182	C₃×M₁₆	C₂×D₄	1	16	yes	complemented
183	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
184	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
185	C₂×C₃×C₈	C₂×D₄	1	8	yes	—
186	C₂×C₃×C₈	C₂×D₄	1	16	yes	complemented
187	C₂×C₃×C₈	C₂⁴	1	2	yes	—
188	C₂×C₃×C₈	C₂⁴	1	4	yes	—
189	C₂×C₃×C₈	C₂⁴	1	2	yes	—
190	C₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
191	C₃×C₂²⋊C₄	—	2	32	no	—
192	C₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
193	C₃×C₄²	C₂×D₄	1	4	yes	—
194	C₃×C₄²	C₂⁴	1	4	yes	—
195	C₂⁵	—	2	2	no	—
196	C₂×#41	—	2	4	no	—
197	C₂×#41	—	2	8	no	—
198	C₂×#41	—	2	8	no	—
199	C₂×#41	C₂³×C₃	1	8	yes	—
200	C₂²×Q₈	C₂³×C₃	1	2	yes	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	—	2	8	no	—
203	C₂²×D₄	—	2	8	no	—
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	—	2	4	no	—
206	C₂²×D₄	—	2	8	no	—
207	C₂²×D₄	C₂³×C₃	1	4	yes	—
208	C₂²×D₄	C₂³×C₃	1	2	yes	—
209	C₂²×D₄	C₂³×C₃	1	8	yes	—
210	C₂³×C₄	—	2	2	no	—
211	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
212	AGL(1,ℤ/8)	—	2	64	no	—
213	#135	—	2	32	no	—
214	C₂×Q₁₆	—	2	4	no	—
215	C₂×Q₁₆	C₂³×C₃	1	4	yes	complemented
216	C₂×SD₁₆	—	2	8	no	—
217	C₂×SD₁₆	—	2	16	no	—
218	C₂×SD₁₆	C₂³×C₃	1	8	yes	complemented
219	C₂×SD₁₆	C₂³×C₃	1	16	yes	complemented
220	C₂×D₈	—	2	8	no	—
221	C₂×D₈	—	2	8	no	—
222	C₂×D₈	—	2	4	no	—
223	C₂×D₈	—	2	16	no	—
224	C₂×D₈	—	2	32	no	—
225	C₂×D₈	C₂³×C₃	1	8	yes	—
226	C₂×D₈	C₂³×C₃	1	8	yes	—
227	C₂×D₈	C₂³×C₃	1	4	yes	complemented
228	C₂×D₈	C₂³×C₃	1	16	yes	complemented
229	C₂×M₁₆	C₃×D₄	1	8	yes	complemented
230	C₂×M₁₆	C₂³×C₃	1	4	yes	—
231	C₂²×C₈	C₃×D₄	1	8	yes	complemented
232	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
234	C₂²×C₈	C₂³×C₃	1	2	yes	—
235	C₂²×C₈	C₂³×C₃	1	4	yes	—
236	#127	—	2	8	no	—
237	#127	C₂³×C₃	1	16	yes	complemented
238	#124	—	2	8	no	—
239	#124	C₂³×C₃	1	16	yes	complemented
240	#121	—	2	32	no	—
241	#120	—	2	32	no	—
242	#117	C₃×D₄	1	4	yes	—
243	#117	C₂³×C₃	1	4	yes	—
244	C₂×C₄⋊C₄	C₃×D₄	1	2	yes	—
245	C₂×C₂²⋊C₄	—	2	8	no	—
246	C₂×C₂²⋊C₄	C₃×D₄	1	2	yes	—
247	C₂×C₄²	C₂³×C₃	1	2	yes	—
248	#97	C₂³×C₃	1	8	yes	complemented
249	C₄×C₈	C₂³×C₃	1	8	yes	complemented
250	C₂³×C₃	—	2	4	no	—
251	C₂³×C₃	—	2	8	no	—
252	C₂³×C₃	—	2	4	no	—
253	C₂³×C₃	—	4	4	no	—
254	C₂³×C₃	—	4	16	no	—
255	C₂³×C₃	—	4	8	no	—
256	C₂³×C₃	—	4	8	no	—
257	C₂³×C₃	—	4	16	no	—
258	C₂³×C₃	—	8	16	no	—
259	C₂³×C₃	—	8	32	no	—
260	C₂³×C₃	#127	1	4	yes	—
261	C₂³×C₃	C₂²×D₄	1	2	yes	—
262	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
263	C₃×Q₈	—	2	4	no	—
264	C₃×Q₈	—	2	4	no	—
265	C₃×Q₈	—	4	4	no	—
266	C₃×D₄	—	2	8	no	—
267	C₃×D₄	—	2	4	no	—
268	C₃×D₄	—	2	8	no	—
269	C₃×D₄	—	2	4	no	—
270	C₃×D₄	—	4	16	no	—
271	C₃×D₄	—	4	16	no	—
272	C₃×D₄	—	4	8	no	—
273	C₃×D₄	—	4	4	no	—
274	C₃×D₄	—	8	32	no	—
275	C₂×C₃×C₄	—	2	8	no	—
276	C₂×C₃×C₄	—	2	4	no	—
277	C₂×C₃×C₄	—	4	4	no	—
278	C₂×C₃×C₄	—	4	16	no	—
279	C₂×C₃×C₄	#127	1	4	yes	—
280	C₂×C₃×C₄	#127	1	8	yes	—
281	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
282	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
284	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
285	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
286	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
287	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
288	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
289	C₃×C₈	C₂²×D₄	1	4	yes	—
290	C₃×C₈	C₂²×D₄	1	4	yes	—
291	C₃×C₈	C₂²×D₄	1	8	yes	—
292	C₂⁴	—	2	4	no	—
293	C₂⁴	—	2	2	no	—
294	C₂⁴	—	2	8	no	—
295	C₂⁴	—	4	8	no	—
296	C₂⁴	—	8	8	no	—
297	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
298	#41	—	2	16	no	—
299	#41	—	2	16	no	—
300	#41	—	4	16	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	2	4	no	—
304	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
305	C₂×D₄	—	2	8	no	—
306	C₂×D₄	—	2	8	no	—
307	C₂×D₄	—	2	16	no	—
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	16	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	4	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	32	no	—
316	C₂×D₄	—	4	8	no	—
317	C₂×D₄	—	4	8	no	—
318	C₂×D₄	—	4	16	no	—
319	C₂×D₄	—	8	16	no	—
320	C₂×D₄	C₂⁴×C₃	1	8	yes	—
321	C₂×D₄	C₂⁴×C₃	1	4	yes	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	4	8	no	—
325	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
326	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
327	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
328	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
329	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
330	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
331	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
332	Q₁₆	—	2	8	no	—
333	SD₁₆	—	2	16	no	—
334	SD₁₆	—	2	32	no	—
335	D₈	—	2	16	no	—
336	D₈	—	2	16	no	—
337	D₈	—	2	8	no	—
338	D₈	—	2	32	no	—
339	M₁₆	C₂×C₃×D₄	1	16	yes	complemented
340	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
341	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
342	C₂×C₈	C₂×C₃×D₄	1	8	yes	—
343	C₂×C₈	C₂×C₃×D₄	1	16	yes	complemented
344	C₂×C₈	C₂⁴×C₃	1	2	yes	—
345	C₂×C₈	C₂⁴×C₃	1	4	yes	—
346	C₂×C₈	C₂⁴×C₃	1	2	yes	—
347	C₄⋊C₄	C₂×C₃×D₄	1	8	yes	—
348	C₂²⋊C₄	—	2	32	no	—
349	C₂²⋊C₄	C₂×C₃×D₄	1	8	yes	—
350	C₄²	C₂×C₃×D₄	1	4	yes	—
351	C₄²	C₂⁴×C₃	1	4	yes	—
352	C₂²×C₃	—	2	8	no	—
353	C₂²×C₃	—	2	4	no	—
354	C₂²×C₃	—	4	4	no	—
355	C₂²×C₃	—	4	8	no	—
356	C₂²×C₃	—	4	8	no	—
357	C₂²×C₃	—	4	16	no	—
358	C₂²×C₃	—	8	32	no	—
359	C₂²×C₃	—	8	8	no	—
360	C₂²×C₃	—	8	16	no	—
361	C₂²×C₃	C₂×#127	1	4	yes	—
362	C₂²×C₃	C₂×#127	1	2	yes	—
363	C₂²×C₃	#576	1	4	yes	—
364	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
365	C₃×C₄	—	2	8	no	—
366	C₃×C₄	—	4	4	no	—
367	C₃×C₄	C₂×#127	1	4	yes	—
368	C₃×C₄	C₂×#127	1	2	yes	—
369	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
370	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
371	C₂³	—	2	4	no	—
372	C₂³	—	2	8	no	—
373	C₂³	—	2	4	no	—
374	C₂³	—	4	8	no	—
375	C₂³	—	4	16	no	—
376	C₂³	—	4	4	no	—
377	C₂³	—	4	16	no	—
378	C₂³	—	4	8	no	—
379	C₂³	—	8	16	no	—
380	C₂³	—	8	32	no	—
381	C₂³	C₃×#127	1	4	yes	—
382	C₂³	C₂²×C₃×D₄	1	2	yes	—
383	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	4	no	—
386	Q₈	—	4	4	no	—
387	D₄	—	2	8	no	—
388	D₄	—	2	4	no	—
389	D₄	—	2	8	no	—
390	D₄	—	2	4	no	—
391	D₄	—	4	16	no	—
392	D₄	—	4	16	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	4	no	—
395	D₄	—	8	32	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	16	no	—
400	C₂×C₄	C₃×#127	1	4	yes	—
401	C₂×C₄	C₃×#127	1	8	yes	—
402	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
403	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
404	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
405	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
406	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
407	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
408	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
409	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
410	C₈	C₂²×C₃×D₄	1	4	yes	—
411	C₈	C₂²×C₃×D₄	1	4	yes	—
412	C₈	C₂²×C₃×D₄	1	8	yes	—
413	C₂×C₃	—	2	4	no	—
414	C₂×C₃	—	8	4	no	—
415	C₂×C₃	—	8	8	no	—
416	C₂×C₃	#3155	1	4	yes	direct product factor
417	C₂×C₃	C₂²×#127	1	1	yes	characteristic
418	C₂×C₃	C₂×#576	1	2	yes	—
419	C₂²	—	2	8	no	—
420	C₂²	—	2	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	8	no	—
423	C₂²	—	4	8	no	—
424	C₂²	—	4	16	no	—
425	C₂²	—	8	32	no	—
426	C₂²	—	8	16	no	—
427	C₂²	—	8	8	no	—
428	C₂²	C₂×C₃×#127	1	4	yes	—
429	C₂²	C₂×C₃×#127	1	2	yes	—
430	C₂²	C₃×#576	1	4	yes	—
431	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
432	C₄	—	2	8	no	—
433	C₄	—	4	4	no	—
434	C₄	C₂×C₃×#127	1	4	yes	—
435	C₄	C₂×C₃×#127	1	2	yes	—
436	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
437	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
438	C₃	C₂×#3155	1	1	yes	characteristic, direct product factor, Sylow
439	C₂	—	2	4	no	—
440	C₂	—	8	8	no	—
441	C₂	—	8	4	no	—
442	C₂	C₃×#3155	1	4	yes	direct product factor
443	C₂	C₂²×C₃×#127	1	1	yes	characteristic
444	C₂	C₂×C₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×C₃×#135	C₂	semidirect
3. C₂³×C₃×D₈	C₂	semidirect
4. C₂×C₃×#535	C₂	semidirect
5. C₃×#3155	C₂	direct (N × H)
6. C₂×C₃×#496	C₂	semidirect
7. C₂×C₃×#495	C₂	semidirect
8. C₂×C₃×#493	C₂	semidirect
9. C₂×C₃×#405	C₂	semidirect
10. C₂×#3155	C₃	direct (N × H)
14. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₃×#135	C₂²	semidirect
17. C₂²×C₃×Q₁₆	C₂²	semidirect
18. C₂²×C₃×SD₁₆	C₂²	semidirect
19. C₂²×C₃×SD₁₆	C₂²	semidirect
23. C₂²×C₃×D₈	C₂²	semidirect
24. C₂²×C₃×D₈	C₂²	semidirect
25. C₂²×C₃×D₈	C₂²	semidirect
29. C₃×#535	C₂²	semidirect
30. C₂×C₃×#127	C₂²	semidirect
31. C₂×C₃×#124	C₂²	semidirect
35. C₃×#496	C₂²	semidirect
36. C₃×#495	C₂²	semidirect
37. C₃×#493	C₂²	semidirect
38. C₃×#405	C₂²	semidirect
39. C₂×C₃×#97	C₂²	semidirect
40. C₂×C₃×C₄×C₈	C₂²	semidirect
41. C₂²×AGL(1,ℤ/8)	C₂×C₃	semidirect
42. C₂²×#135	C₂×C₃	semidirect
43. C₂³×D₈	C₂×C₃	semidirect
44. C₂×#535	C₂×C₃	semidirect
45. #3155	C₂×C₃	direct (N × H)
46. C₂×#496	C₂×C₃	semidirect
47. C₂×#495	C₂×C₃	semidirect
48. C₂×#493	C₂×C₃	semidirect
49. C₂×#405	C₂×C₃	semidirect
70. C₂×C₃×Q₁₆	C₂³	semidirect
73. C₂×C₃×SD₁₆	C₂³	semidirect
74. C₂×C₃×SD₁₆	C₂³	semidirect
82. C₂×C₃×D₈	C₂³	semidirect
83. C₂×C₃×D₈	C₂³	semidirect
84. C₂×C₃×M₁₆	D₄	semidirect
86. C₂²×C₃×C₈	D₄	semidirect
92. C₃×#127	C₂³	semidirect
94. C₃×#124	C₂³	semidirect
103. C₃×#97	C₂³	semidirect
104. C₃×C₄×C₈	C₂³	semidirect
108. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
110. C₂×#135	C₂²×C₃	semidirect
111. C₂²×Q₁₆	C₂²×C₃	semidirect
112. C₂²×SD₁₆	C₂²×C₃	semidirect
113. C₂²×SD₁₆	C₂²×C₃	semidirect
117. C₂²×D₈	C₂²×C₃	semidirect
118. C₂²×D₈	C₂²×C₃	semidirect
119. C₂²×D₈	C₂²×C₃	semidirect
123. #535	C₂²×C₃	semidirect
124. C₂×#127	C₂²×C₃	semidirect
125. C₂×#124	C₂²×C₃	semidirect
129. #496	C₂²×C₃	semidirect
130. #495	C₂²×C₃	semidirect
131. #493	C₂²×C₃	semidirect
132. #405	C₂²×C₃	semidirect
133. C₂×#97	C₂²×C₃	semidirect
134. C₂×C₄×C₈	C₂²×C₃	semidirect
182. C₃×M₁₆	C₂×D₄	semidirect
186. C₂×C₃×C₈	C₂×D₄	semidirect
215. C₂×Q₁₆	C₂³×C₃	semidirect
218. C₂×SD₁₆	C₂³×C₃	semidirect
219. C₂×SD₁₆	C₂³×C₃	semidirect
227. C₂×D₈	C₂³×C₃	semidirect
228. C₂×D₈	C₂³×C₃	semidirect
229. C₂×M₁₆	C₃×D₄	semidirect
231. C₂²×C₈	C₃×D₄	semidirect
237. #127	C₂³×C₃	semidirect
239. #124	C₂³×C₃	semidirect
248. #97	C₂³×C₃	semidirect
249. C₄×C₈	C₂³×C₃	semidirect
339. M₁₆	C₂×C₃×D₄	semidirect
343. C₂×C₈	C₂×C₃×D₄	semidirect
416. C₂×C₃	#3155	direct (N × H)
438. C₃	C₂×#3155	direct (N × H)
442. C₂	C₃×#3155	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×AGL(1,ℤ/8)	2
C₂²×C₃×#135	2
C₂³×C₃×D₈	2
C₂×C₃×#535	2
C₃×#3155	2
C₂×C₃×#496	2
C₂×C₃×#495	2
C₂×C₃×#493	2
C₂×C₃×#405	2
C₂×#3155	3

Subgroups — C2×C3×#3155 #10679248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3155 #10679248